Question 5
Let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be $2n$ positive integers such that the $n+1$ products
$$
\begin{gathered}
a_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} b_{2} a_{3} \cdots a_{n}, \\
\vdots \\
b_{1} b_{2} b_{3} \cdots b_{n}
\end{gathered}
$$
form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.
$$
\begin{gathered}
a_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} b_{2} a_{3} \cdots a_{n}, \\
\vdots \\
b_{1} b_{2} b_{3} \cdots b_{n}
\end{gathered}
$$
form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.